Question: You have found the following ages (in years) of 6 snakes. Those snakes were randomly selected from the 31 snakes at your local zoo: $ 20,\enspace 20,\enspace 3,\enspace 14,\enspace 14,\enspace 8$ Based on your sample, what is the average age of the snakes? What is the standard deviation? You may round your answers to the nearest tenth.
Explanation: Because we only have data for a small sample of the 31 snakes, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{20 + 20 + 3 + 14 + 14 + 8}{{6}} = {13.2\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {46.24} + {46.24} + {104.04} + {0.64} + {0.64} + {27.04}} {{6 - 1}} $ {s^2} = \dfrac{{224.84}}{{5}} = {44.97\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{44.97\text{ years}^2}} = {6.7\text{ years}} $ We can estimate that the average snake at the zoo is 13.2 years old. There is also a standard deviation of 6.7 years.